Polina,
Could you please also confirm if the Hosts were successfully upd ated but the Viewer was showing the wrong version?
Unable to confirm. I thought I upgraded each client in two offices to 6.12.1. Today, I was onsite at one of the offices and confirmed seeing 6.10.10 running on a specific host (so my initial response would have been "no, they didn't actually get upd ated"). I got home, did the advanced remote install again (technically, on a different PC than I did before), waited 3 minutes and reconnected. The host reported 6.12.1. So, not reproduced. I'll see if I have any more hosts I can try.
Could you please let me know if you have the Use legacy capture mode feature enabled in the Settings for Host -> Other tab? If not, could you please try enabling it and see if it helps to resolve the black screen issue?
Yes, that works. It was unchecked by default. After enabling it, I was able to connect and type into the password field to login. I closed and reconnected a couple of times, haven't seen the black screen come back and the mouse/KB have been responsive since enabling Legacy mode.
However, it didn't resolve the weird mouse offset focus problem. In the lower right corner I right clicked on the RUT hosts icon, and the menu opened up on the far left side! I've now observed this on multiple hosts, both 6.10.10 and 6.12.1. Previously, I was clicking the middle or bottom middle of the screen and seeing mouse act like it was coming from the left. Today was the far right click that still behaved like it came from the left side of the screen.

Please try referring to this KB article regarding the troubleshooting of the slow remote connection speed. If you have already tried the solutions listed there but none of them helped to resolve the issue, then you might send us the Host log files for the examination.
Note, the connection isn't slow, just the time to establish it in the first place is slow, in my opinion. I think its the authorization being set to "auto". By default, they are se t to auto. When I connected to 3 hosts, I could see it do the first of 5 attempts, fail and succeed on the next attempt. This all takes like 12-13 seconds or more.
On the 4 host, I just went and changed it from Auto to single password mode, and it seemed to take much less time, like 5 seconds instead of 15. It didn't show me any window about trying the first of 5 attempts and failing on the first try. I just se t another host to use single password instead of auto. It took about 12 seconds for the remote desktop to appear. It feels like it saved a few seconds, but not enough cycles to know if that is consistent or just my brain getting fooled.
I'll see about getting some Viewer logs to see if there is any time being wasted that can be avoided.